Hamilton almost doubled their lead a minute from the break when McGowan tried an audacious chip that rebounded off the crossbar.
Morrison forced a fine save from Cerny in the first minute of the second half when he broke free inside the Accies box.
Richard Offiong burst clear at the other end, but he ran the ball out before he could get a shot on target.
However, County looked to finally be getting a grip on the tie and were seeing a lot more of the ball.
Accies were awarded a free-kick in the 58th minute, but Easton curled the ball wide of goalkeeper Derek Soutar's left-hand post.
County missed a great chance in the 65th minute when Richie Hart failed to get enough power in his header right in front of goal after Steven Watt supplied an inch-perfect cross.
Accies opened up the County defence in the 74th minute when McGowan crossed into the box, but substitute Derek Lyle failed to connect with the goal gaping.
County brought on new signing Stewart in an effort to pep up their attack, but he saw little of the ball.
Stuart Golabek tried a long-range effort from 30 yards, but his shot flew narrowly wide of Cerny's right-hand post.
Accies had McArthur sent off in the final minute when he picked up a second booking and, seconds later, Craig fired into the side-netting.
Ross County manager Derek Adams: "We're majorly disappointed. I just feel that two decisions from the standside linesman and referee Crawford Allan have cost us the game.
"He should have sent their goalkeeper off - it looked like it was a handball outside the penalty area - and James McArthur should have been sent off for an intentional lunge at one of our players early in the first half.
"That would have changed the game, these decisions are massive, and they didn't go for us. I'm very disappointed and angry because that affects people's livelihoods.
"Hamilton have gone through but we've performed well, totally dominated the second half. The SFA ask us for respect? Well, they'll actually have to improve their performance ... the referee's put us out the cup."
Hamilton manager Billy Reid: "We knew Ross County were a good side and have plenty of players with SPL experience, conditions were poor and we're just delighted to be in the next round.
"We said that we had a height advantage over Ross County. Paul McGowan put in a smashing cross for Chris Swailes to nod home and that was very, very important.
"We almost added a second through McGowan's chip that hit the crossbar. I thought we controlled the first 60 minutes of the game, but Ross County came into it in the last 30.
"I thought the sending-off of James McArthur was harsh, but defensively we were excellent."
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